## Chaos on the Circle |

**Professor Emeritus Susan Niefield**

Union College

February 8, 2016

5:00PM

Bailey Hall 207

Refreshments will be served in Bailey Hall 204 4:45pm

Take a point $P$ on a circle in the plane and double the angle $\theta$ to get a point $f(P)$.Repeating this process gives a set $\{P,f(P),f(f(P)),\dots \}$ of points, called the

orbitof $P$ under $f$}. What kind of orbits can we find? Of course, that depends on the starting point $P$. Some orbits are finite, while others are not. Among the infinite ones, there are even orbits that hit almost every point on the circle.This map is an example of a

chaotic dynamical system. After presenting a definition ofchaos, we will show that points on a circle can be represented as binary sequences, and use this representation to prove that the angle-doubling map $f$ is chaotic.

For additional information, send e-mail to math@union.edu or call (518) 388-6246.

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